Introduction to Stochastic Calculus Math 545
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A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation
fractal and fractional Article A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation Marylu L. Lagunes, Oscar Castillo * , Fevrier Valdez , Jose Soria and Patricia Melin Tijuana Institute of Technology, Calzada Tecnologico s/n, Fracc. Tomas Aquino, Tijuana 22379, Mexico; [email protected] (M.L.L.); [email protected] (F.V.); [email protected] (J.S.); [email protected] (P.M.) * Correspondence: [email protected] Abstract: Stochastic fractal search (SFS) is a novel method inspired by the process of stochastic growth in nature and the use of the fractal mathematical concept. Considering the chaotic stochastic diffusion property, an improved dynamic stochastic fractal search (DSFS) optimization algorithm is presented. The DSFS algorithm was tested with benchmark functions, such as the multimodal, hybrid, and composite functions, to evaluate the performance of the algorithm with dynamic parameter adaptation with type-1 and type-2 fuzzy inference models. The main contribution of the article is the utilization of fuzzy logic in the adaptation of the diffusion parameter in a dynamic fashion. This parameter is in charge of creating new fractal particles, and the diversity and iteration are the input information used in the fuzzy system to control the values of diffusion. Keywords: fractal search; fuzzy logic; parameter adaptation; CEC 2017 Citation: Lagunes, M.L.; Castillo, O.; Valdez, F.; Soria, J.; Melin, P. A New Approach for Dynamic Stochastic 1. Introduction Fractal Search with Fuzzy Logic for Metaheuristic algorithms are applied to optimization problems due to their charac- Parameter Adaptation. Fractal Fract. teristics that help in searching for the global optimum, while simple heuristics are mostly 2021, 5, 33. -
Introduction to Stochastic Processes - Lecture Notes (With 33 Illustrations)
Introduction to Stochastic Processes - Lecture Notes (with 33 illustrations) Gordan Žitković Department of Mathematics The University of Texas at Austin Contents 1 Probability review 4 1.1 Random variables . 4 1.2 Countable sets . 5 1.3 Discrete random variables . 5 1.4 Expectation . 7 1.5 Events and probability . 8 1.6 Dependence and independence . 9 1.7 Conditional probability . 10 1.8 Examples . 12 2 Mathematica in 15 min 15 2.1 Basic Syntax . 15 2.2 Numerical Approximation . 16 2.3 Expression Manipulation . 16 2.4 Lists and Functions . 17 2.5 Linear Algebra . 19 2.6 Predefined Constants . 20 2.7 Calculus . 20 2.8 Solving Equations . 22 2.9 Graphics . 22 2.10 Probability Distributions and Simulation . 23 2.11 Help Commands . 24 2.12 Common Mistakes . 25 3 Stochastic Processes 26 3.1 The canonical probability space . 27 3.2 Constructing the Random Walk . 28 3.3 Simulation . 29 3.3.1 Random number generation . 29 3.3.2 Simulation of Random Variables . 30 3.4 Monte Carlo Integration . 33 4 The Simple Random Walk 35 4.1 Construction . 35 4.2 The maximum . 36 1 CONTENTS 5 Generating functions 40 5.1 Definition and first properties . 40 5.2 Convolution and moments . 42 5.3 Random sums and Wald’s identity . 44 6 Random walks - advanced methods 48 6.1 Stopping times . 48 6.2 Wald’s identity II . 50 6.3 The distribution of the first hitting time T1 .......................... 52 6.3.1 A recursive formula . 52 6.3.2 Generating-function approach . -
1 Stochastic Processes and Their Classification
1 1 STOCHASTIC PROCESSES AND THEIR CLASSIFICATION 1.1 DEFINITION AND EXAMPLES Definition 1. Stochastic process or random process is a collection of random variables ordered by an index set. ☛ Example 1. Random variables X0;X1;X2;::: form a stochastic process ordered by the discrete index set f0; 1; 2;::: g: Notation: fXn : n = 0; 1; 2;::: g: ☛ Example 2. Stochastic process fYt : t ¸ 0g: with continuous index set ft : t ¸ 0g: The indices n and t are often referred to as "time", so that Xn is a descrete-time process and Yt is a continuous-time process. Convention: the index set of a stochastic process is always infinite. The range (possible values) of the random variables in a stochastic process is called the state space of the process. We consider both discrete-state and continuous-state processes. Further examples: ☛ Example 3. fXn : n = 0; 1; 2;::: g; where the state space of Xn is f0; 1; 2; 3; 4g representing which of four types of transactions a person submits to an on-line data- base service, and time n corresponds to the number of transactions submitted. ☛ Example 4. fXn : n = 0; 1; 2;::: g; where the state space of Xn is f1; 2g re- presenting whether an electronic component is acceptable or defective, and time n corresponds to the number of components produced. ☛ Example 5. fYt : t ¸ 0g; where the state space of Yt is f0; 1; 2;::: g representing the number of accidents that have occurred at an intersection, and time t corresponds to weeks. ☛ Example 6. fYt : t ¸ 0g; where the state space of Yt is f0; 1; 2; : : : ; sg representing the number of copies of a software product in inventory, and time t corresponds to days. -
Monte Carlo Sampling Methods
[1] Monte Carlo Sampling Methods Jasmina L. Vujic Nuclear Engineering Department University of California, Berkeley Email: [email protected] phone: (510) 643-8085 fax: (510) 643-9685 UCBNE, J. Vujic [2] Monte Carlo Monte Carlo is a computational technique based on constructing a random process for a problem and carrying out a NUMERICAL EXPERIMENT by N-fold sampling from a random sequence of numbers with a PRESCRIBED probability distribution. x - random variable N 1 xˆ = ---- x N∑ i i = 1 Xˆ - the estimated or sample mean of x x - the expectation or true mean value of x If a problem can be given a PROBABILISTIC interpretation, then it can be modeled using RANDOM NUMBERS. UCBNE, J. Vujic [3] Monte Carlo • Monte Carlo techniques came from the complicated diffusion problems that were encountered in the early work on atomic energy. • 1772 Compte de Bufon - earliest documented use of random sampling to solve a mathematical problem. • 1786 Laplace suggested that π could be evaluated by random sampling. • Lord Kelvin used random sampling to aid in evaluating time integrals associated with the kinetic theory of gases. • Enrico Fermi was among the first to apply random sampling methods to study neutron moderation in Rome. • 1947 Fermi, John von Neuman, Stan Frankel, Nicholas Metropolis, Stan Ulam and others developed computer-oriented Monte Carlo methods at Los Alamos to trace neutrons through fissionable materials UCBNE, J. Vujic Monte Carlo [4] Monte Carlo methods can be used to solve: a) The problems that are stochastic (probabilistic) by nature: - particle transport, - telephone and other communication systems, - population studies based on the statistics of survival and reproduction. -
Chaotic Decompositions in Z2-Graded Quantum Stochastic Calculus
QUANTUM PROBABILITY BANACH CENTER PUBLICATIONS, VOLUME 43 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1998 CHAOTIC DECOMPOSITIONS IN Z2-GRADED QUANTUM STOCHASTIC CALCULUS TIMOTHY M. W. EYRE Department of Mathematics, University of Nottingham Nottingham, NG7 2RD, UK E-mail: [email protected] Abstract. A brief introduction to Z2-graded quantum stochastic calculus is given. By inducing a superalgebraic structure on the space of iterated integrals and using the heuristic classical relation df(Λ) = f(Λ+dΛ)−f(Λ) we provide an explicit formula for chaotic expansions of polynomials of the integrator processes of Z2-graded quantum stochastic calculus. 1. Introduction. A theory of Z2-graded quantum stochastic calculus was was in- troduced in [EH] as a generalisation of the one-dimensional Boson-Fermion unification result of quantum stochastic calculus given in [HP2]. Of particular interest in Z2-graded quantum stochastic calculus is the result that the integrators of the theory provide a time-indexed family of representations of a broad class of Lie superalgebras. The notion of a Lie superalgebra was introduced in [K] and has received considerable attention since. It is essentially a Z2-graded analogue of the notion of a Lie algebra with a bracket that is, in a certain sense, partly a commutator and partly an anticommutator. Another work on Lie superalgebras is [S] and general superalgebras are treated in [C,S]. The Lie algebra representation properties of ungraded quantum stochastic calculus enabled an explicit formula for the chaotic expansion of elements of an associated uni- versal enveloping algebra to be developed in [HPu]. -
A Stochastic Fractional Calculus with Applications to Variational Principles
fractal and fractional Article A Stochastic Fractional Calculus with Applications to Variational Principles Houssine Zine †,‡ and Delfim F. M. Torres *,‡ Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal; [email protected] * Correspondence: delfi[email protected] † This research is part of first author’s Ph.D. project, which is carried out at the University of Aveiro under the Doctoral Program in Applied Mathematics of Universities of Minho, Aveiro, and Porto (MAP). ‡ These authors contributed equally to this work. Received: 19 May 2020; Accepted: 30 July 2020; Published: 1 August 2020 Abstract: We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation. Keywords: fractional derivatives and integrals; stochastic processes; calculus of variations MSC: 26A33; 49K05; 60H10 1. Introduction A stochastic calculus of variations, which generalizes the ordinary calculus of variations to stochastic processes, was introduced in 1981 by Yasue, generalizing the Euler–Lagrange equation and giving interesting applications to quantum mechanics [1]. Recently, stochastic variational differential equations have been analyzed for modeling infectious diseases [2,3], and stochastic processes have shown to be increasingly important in optimization [4]. In 1996, fifteen years after Yasue’s pioneer work [1], the theory of the calculus of variations evolved in order to include fractional operators and better describe non-conservative systems in mechanics [5]. -
A Short History of Stochastic Integration and Mathematical Finance
A Festschrift for Herman Rubin Institute of Mathematical Statistics Lecture Notes – Monograph Series Vol. 45 (2004) 75–91 c Institute of Mathematical Statistics, 2004 A short history of stochastic integration and mathematical finance: The early years, 1880–1970 Robert Jarrow1 and Philip Protter∗1 Cornell University Abstract: We present a history of the development of the theory of Stochastic Integration, starting from its roots with Brownian motion, up to the introduc- tion of semimartingales and the independence of the theory from an underlying Markov process framework. We show how the development has influenced and in turn been influenced by the development of Mathematical Finance Theory. The calendar period is from 1880 to 1970. The history of stochastic integration and the modelling of risky asset prices both begin with Brownian motion, so let us begin there too. The earliest attempts to model Brownian motion mathematically can be traced to three sources, each of which knew nothing about the others: the first was that of T. N. Thiele of Copen- hagen, who effectively created a model of Brownian motion while studying time series in 1880 [81].2; the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900 (see, [1, 2, 11]); and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matter, in 1905 [21](See [64] for a discussion of Einstein’s model and his motivations.) Of these three models, those of Thiele and Bachelier had little impact for a long time, while that of Einstein was immediately influential. -
Informal Introduction to Stochastic Calculus 1
MATH 425 PART V: INFORMAL INTRODUCTION TO STOCHASTIC CALCULUS G. BERKOLAIKO 1. Motivation: how to model price paths Differential equations have long been an efficient tool to model evolution of various mechanical systems. At the start of 20th century, however, science started to tackle modeling of systems driven by probabilistic effects, such as stock market (the work of Bachelier) or motion of small particles suspended in liquid (the work of Einstein and Smoluchowski). For us, the motivating questions are: Is there an equation describing the evolution of a stock price? And, perhaps more applicably, how can we simulate numerically a seemingly random path taken by the stock price? Example 1.1. We have seen in previous sections that we can model the distribution of stock prices at time t as p 2 St = S0 exp (µ − σ =2)t + σ tY ; (1.1) where Y is standard normal random variable. So perhaps, for every time t we can generate a standard normal variable, substitute it into equation (1.1) and plot the resulting St as a function of t? The result is shown in Figure 1 and it definitely does not look like a price path of a stock. The underlying reason is that we used equation (1.1) to simulate prices at different time points independently. However, if t1 is close to t2, the prices cannot be completely independent. It is the change in price that we assumed (in “Efficient Market Hypothesis") to be independent of the previous price changes. 2. Wiener process Definition 2.1. Wiener process (or Brownian motion) is a random function of t, denoted Wt = Wt(!) (where ! is the underlying random event) such that (a) Wt is a.s. -
Lecture Notes on Stochastic Calculus (Part II)
Lecture Notes on Stochastic Calculus (Part II) Fabrizio Gelsomino, Olivier L´ev^eque,EPFL July 21, 2009 Contents 1 Stochastic integrals 3 1.1 Ito's integral with respect to the standard Brownian motion . 3 1.2 Wiener's integral . 4 1.3 Ito's integral with respect to a martingale . 4 2 Ito-Doeblin's formula(s) 7 2.1 First formulations . 7 2.2 Generalizations . 7 2.3 Continuous semi-martingales . 8 2.4 Integration by parts formula . 9 2.5 Back to Fisk-Stratonoviˇc'sintegral . 10 3 Stochastic differential equations (SDE's) 11 3.1 Reminder on ordinary differential equations (ODE's) . 11 3.2 Time-homogeneous SDE's . 11 3.3 Time-inhomogeneous SDE's . 13 3.4 Weak solutions . 14 4 Change of probability measure 15 4.1 Exponential martingale . 15 4.2 Change of probability measure . 15 4.3 Martingales under P and martingales under PeT ........................ 16 4.4 Girsanov's theorem . 17 4.5 First application to SDE's . 18 4.6 Second application to SDE's . 19 4.7 A particular case: the Black-Scholes model . 20 4.8 Application : pricing of a European call option (Black-Scholes formula) . 21 1 5 Relation between SDE's and PDE's 23 5.1 Forward PDE . 23 5.2 Backward PDE . 24 5.3 Generator of a diffusion . 26 5.4 Markov property . 27 5.5 Application: option pricing and hedging . 28 6 Multidimensional processes 30 6.1 Multidimensional Ito-Doeblin's formula . 30 6.2 Multidimensional SDE's . 31 6.3 Drift vector, diffusion matrix and weak solution . -
Random Numbers and Stochastic Simulation
Stochastic Simulation and Randomness Random Number Generators Quasi-Random Sequences Scientific Computing: An Introductory Survey Chapter 13 – Random Numbers and Stochastic Simulation Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 17 Stochastic Simulation and Randomness Random Number Generators Quasi-Random Sequences Stochastic Simulation Stochastic simulation mimics or replicates behavior of system by exploiting randomness to obtain statistical sample of possible outcomes Because of randomness involved, simulation methods are also known as Monte Carlo methods Such methods are useful for studying Nondeterministic (stochastic) processes Deterministic systems that are too complicated to model analytically Deterministic problems whose high dimensionality makes standard discretizations infeasible (e.g., Monte Carlo integration) < interactive example > < interactive example > Michael T. Heath Scientific Computing 2 / 17 Stochastic Simulation and Randomness Random Number Generators Quasi-Random Sequences Stochastic Simulation, continued Two main requirements for using stochastic simulation methods are Knowledge of relevant probability distributions Supply of random numbers for making random choices Knowledge of relevant probability distributions depends on theoretical or empirical information about physical system being simulated By simulating large number of trials, probability -
Introduction to Stochastic Calculus
Introduction Conditional Expectation Martingales Brownian motion Stochastic integral Ito formula Introduction to Stochastic Calculus Rajeeva L. Karandikar Director Chennai Mathematical Institute [email protected] [email protected] Rajeeva L. Karandikar Director, Chennai Mathematical Institute Introduction to Stochastic Calculus - 1 Introduction Conditional Expectation Martingales Brownian motion Stochastic integral Ito formula A Game Consider a gambling house. A fair coin is going to be tossed twice. A gambler has a choice of betting at two games: (i) A bet of Rs 1000 on first game yields Rs 1600 if the first toss results in a Head (H) and yields Rs 100 if the first toss results in Tail (T). (ii) A bet of Rs 1000 on the second game yields Rs 1800, Rs 300, Rs 50 and Rs 1450 if the two toss outcomes are HH,HT,TH,TT respectively. Rajeeva L. Karandikar Director, Chennai Mathematical Institute Introduction to Stochastic Calculus - 2 Introduction Conditional Expectation Martingales Brownian motion Stochastic integral Ito formula Rajeeva L. Karandikar Director, Chennai Mathematical Institute Introduction to Stochastic Calculus - 3 Introduction Conditional Expectation Martingales Brownian motion Stochastic integral Ito formula A Game...... Now it can be seen that the expected return from Game 1 is Rs 850 while from Game 2 is Rs 900 and so on the whole, the gambling house is set to make money if it can get large number of players to play. The novelty of the game (designed by an expert) was lost after some time and the casino owner decided to make it more interesting: he allowed people to bet without deciding if it is for Game 1 or Game 2 and they could decide to stop or continue after first toss. -
Stochastic Calculus of Variations for Stochastic Partial Differential Equations
JOURNAL OF FUNCTIONAL ANALYSIS 79, 288-331 (1988) Stochastic Calculus of Variations for Stochastic Partial Differential Equations DANIEL OCONE* Mathematics Department, Rutgers Universiiy, New Brunswick, New Jersey 08903 Communicated by Paul Malliavin Receved December 1986 This paper develops the stochastic calculus of variations for Hilbert space-valued solutions to stochastic evolution equations whose operators satisfy a coercivity con- dition. An application is made to the solutions of a class of stochastic pde’s which includes the Zakai equation of nonlinear filtering. In particular, a Lie algebraic criterion is presented that implies that all finite-dimensional projections of the solution define random variables which admit a density. This criterion generalizes hypoelhpticity-type conditions for existence and regularity of densities for Iinite- dimensional stochastic differential equations. 0 1988 Academic Press, Inc. 1. INTRODUCTION The stochastic calculus of variation is a calculus for functionals on Wiener space based on a concept of differentiation particularly suited to the invariance properties of Wiener measure. Wiener measure on the space of continuous @-valued paths is quasi-invariant with respect to translation by a path y iff y is absolutely continuous with a square-integrable derivative. Hence, the proper definition of the derivative of a Wiener functional considers variation only in directions of such y. Using this derivative and its higher-order iterates, one can develop Wiener functional Sobolev spaces, which are analogous in most respects to Sobolev spaces of functions over R”, and these spaces provide the right context for discussing “smoothness” and regularity of Wiener functionals. In particular, functionals defined by solving stochastic differential equations driven by a Wiener process are smooth in the stochastic calculus of variations sense; they are not generally smooth in a Frechet sense, which ignores the struc- ture of Wiener measure.